Foundations of Computational Mathematics

, Volume 11, Issue 5, pp 589–616 | Cite as

Representation Theoretic Patterns in Three-Dimensional Cryo-Electron Microscopy II—The Class Averaging Problem

Article

Abstract

In this paper we study the formal algebraic structure underlying the intrinsic classification algorithm, recently introduced in Singer et al. (SIAM J. Imaging Sci. 2011, accepted), for classifying noisy projection images of similar viewing directions in three-dimensional cryo-electron microscopy (cryo-EM). This preliminary classification is of fundamental importance in determining the three-dimensional structure of macromolecules from cryo-EM images. Inspecting this algebraic structure we obtain a conceptual explanation for the admissibility (correctness) of the algorithm and a proof of its numerical stability. The proof relies on studying the spectral properties of an integral operator of geometric origin on the two-dimensional sphere, called the localized parallel transport operator. Along the way, we continue to develop the representation theoretic set-up for three-dimensional cryo-EM that was initiated in Hadani and Singer (Ann. Math. 2010, accepted).

Keywords

Representation theory Differential geometry Spectral theory Optimization theory Mathematical biology 3D cryo-electron microscopy 

Mathematics Subject Classification (2000)

20G05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D.A. Doyle, J.M. Cabral, R.A. Pfuetzner, A. Kuo, J.M. Gulbis, S.L. Cohen, B.T. Chait, R. MacKinnon, The structure of the potassium channel: molecular basis of K+ conduction and selectivity, Science 280, 69–77 (1998). CrossRefGoogle Scholar
  2. 2.
    J. Frank, Three-Dimensional Electron Microscopy of Macromolecular Assemblies. Visualization of Biological Molecules in Their Native State (Oxford Press, Oxford, 2006). CrossRefGoogle Scholar
  3. 3.
    R. Hadani, A. Singer, Representation theoretic patterns in three-dimensional cryo-electron macroscopy I—The Intrinsic reconstitution algorithm, Ann. Math. (2010, accepted). A PDF version can be downloaded from http://www.math.utexas.edu/~hadani.
  4. 4.
    R. MacKinnon, Potassium channels and the atomic basis of selective ion conduction, 8 December 2003, Nobel Lecture, Biosci. Rep. 24(2), 75–100 (2004). CrossRefGoogle Scholar
  5. 5.
    F. Natterer, The Mathematics of Computerized Tomography. Classics in Applied Mathematics (SIAM, Philadelphia, 2001). MATHCrossRefGoogle Scholar
  6. 6.
    P.A. Penczek, J. Zhu, J. Frank, A common-lines based method for determining orientations for N>3 particle projections simultaneously, Ultramicroscopy 63, 205–218 (1996). CrossRefGoogle Scholar
  7. 7.
    A. Singer, Y. Shkolnisky, Three-dimensional structure determination from common lines in cryo-EM by eigenvectors and semidefinite programming, SIAM J. Imaging Sci. (2011, accepted). Google Scholar
  8. 8.
    A. Singer, Z. Zhao, Y. Shkolnisky, R. Hadani, Viewing angle classification of cryo-electron microscopy images using eigenvectors, SIAM J. Imaging Sci. (2011, accepted). A PDF version can be downloaded from http://www.math.utexas.edu/~hadani.
  9. 9.
    G. Szegö, Orthogonal Polynomials. Colloquium Publications, vol. XXIII (American Mathematical Society, Providence, 1939). Google Scholar
  10. 10.
    E.M. Taylor, Noncommutative Harmonic Analysis. Mathematical Surveys and Monographs, vol. 22 (American Mathematical Society, Providence, 1986). MATHGoogle Scholar
  11. 11.
    B. Vainshtein, A. Goncharov, Determination of the spatial orientation of arbitrarily arranged identical particles of an unknown structure from their projections, in Proc. 11th Intern. Congr. on Elec. Mirco (1986), pp. 459–460. Google Scholar
  12. 12.
    M. Van Heel, Angular reconstitution: a posteriori assignment of projection directions for 3D reconstruction, Ultramicroscopy 21(2), 111–123 (1987). PMID: 12425301 [PubMed—indexed for MEDLINE]. CrossRefGoogle Scholar
  13. 13.
    L. Wang, F.J. Sigworth, Cryo-EM and single particles, Plant Physiol. 21, 8–13 (2006). Review. PMID: 16443818 [PubMed—indexed for MEDLINE]. Google Scholar

Copyright information

© SFoCM 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas at AustinAustinUSA
  2. 2.Department of Mathematics and PACMPrinceton UniversityPrincetonUSA

Personalised recommendations